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Take the incomplete metric space $X=(0,1)$ with its usual distance, andEach is nonempty and closed relative to $X$: it is $X\cap (-\infty ,1/(n+1)]$. The sequence is strictly descending and its diameters are $1/(n+1)\to 0$. Its intersection is empty, since every positive point exceeds $1/(n+1)$ for sufficiently large $n$. The excluded limiting point is zero.

A metric space is complete if every Cauchy sequence in it converges to a point of that space. The stated shrinking-set characterization is the Cantor intersection theorem together with its converse.

First assume completeness, and choose $x_n\in A_n$. If $m,n\ge N$, then both points belong to $A_N$, so $d(x_m,x_n)\le \mathrm{diam}(A_N)$. The vanishing diameters imply that $(x_n)$ is a Cauchy sequence, hence it converges to some $x\in X$. For every fixed $N$, its tail belongs to the closed set $A_N$, so $x\in A_N$. Thus $x\in {\bigcap }_N{A}_N$. In fact the intersection contains only one point, since any two of its points have distance at most every diameter.

Conversely assume the stated intersection property and let $(x_n)$ be a Cauchy sequence. PutThese closures are nonempty, closed and nested. Taking a closure does not increase the diameter: approximate two points of the closure by points of the set and use continuity of distance. The Cauchy sequence therefore makes $\mathrm{diam}(A_N)\to 0$. Choose $x$ in the intersection. For $n\ge N$, both $x_n$ and $x$ belong to $A_N$, and $d(x_n,x)\le \mathrm{diam}(A_N)\to 0$. Every Cauchy sequence converges in $X$, proving completeness.

Consider the filled hyperbolic triangle with vertices $A,B,C$. Every point $P$ is on some geodesic segment $Aw$ with $w\in BC$: for instance, in the Beltrami-Klein model these are the ordinary straight segments filling a convex triangle. For any other point $Q$ of the triangle, apply part (b) first on $Aw$ and then on $BC$:Applying the same argument to $Q$, with each vertex held fixed, bounds every term on the right by the maximum of the three side lengths. The endpoints of a longest side attain that bound. Thus the diameter is exactly the longest side length:

For the vertical slit, choose the branch of $\sqrt{z^2+1}$ asymptotic to $z$ at infinity. It is the mapping-out function of a compact H-hull for $(0,i]$, andThus half-plane capacity of a vertical slit givesThe endpoint $0$ is merely part of the boundary closure convention.

For the filled upper half-disc $K=\{z\in \mathbb{H}:|z|\le 1\}$, the conformal map $g_K(z)=z+1/z$ maps the exterior half-disc onto $\mathbb{H}$. Indeed, it sends the semicircle to $[-2,2]$, the remaining real boundary to the complementary intervals, and its inverse is the branch of $(w+\sqrt{w^2-4})/2$ asymptotic to $w$. Hence half-plane capacity of a half-disc givesIf $\mathbb{D}$ denotes the open unit disc, the printed $\mathbb{H}\cap \mathbb{D}$ is not relatively closed and is literally not a compact H-hull. The intended value is the one for its filled relative closure $K$. With a closed-disc convention the displayed notation already represents that hull.

For a nonempty compact H-hull $A$, its closure meets $\mathbb{R}$: otherwise a nonempty compact set strictly inside $\mathbb{H}$ would be separated from the lower half-plane in the complement, contradicting the simply connected domain condition for $\mathbb{H}\setminus A$. Choose $x\in \overline{A}\cap \mathbb{R}$ and put $d=\mathrm{diam}(A)$. Taking limits in the definition of diameter shows $|z-x|\le d$ for every $z\in A$. Thus $A$ lies in the filled half-disc of radius $d$ centered at $x$. Its half-plane capacity is $d^2$ by scaling and translation of half-plane capacity. Using monotonicity of half-plane capacity,For the empty compact H-hull, use diameter zero. This proves the unit-constant version of half-plane capacity is bounded by squared diameter; no claim of optimality of the constant is needed.

Write $g_s=g_{K_s}$ and let $K_{s,t}$ be the relatively closed hull obtained by mapping the increment $K_t\setminus K_s$ through $g_s$. For a bounded compact H-hull $L$, defineThe Loewner local growth property isOne can equivalently use diameter: for a nonempty compact H-hull, $\mathrm{diam}(L)\le 2\mathrm{rad}(L)\le 2\mathrm{diam}(L)$, using a point of its closure on $\mathbb{R}$ for the second inequality. The condition concerns the mapped increment, not the original difference $K_{t+h}\setminus K_t$, which can be large when a narrow opening closes. It makes growth occur near one continuous Loewner driving function rather than by a sudden addition at separated boundary points.

Let $f_t$ be the optical telescope focal length, $x=f_t\theta$ the physical optical slit width, $f_{\mathrm{coll}}$ the collimator focal length, and $A_{\mathrm{coll}}=D{f}_{\mathrm{coll}}/f_t$ the incident collimated beam diameter. For the grating equation $m\lambda =d(\sin \alpha +\sin \beta )$, differentiation at fixed wavelength gives $|d\beta /d\alpha |=\cos \alpha /\cos \beta$. Thus the monochromatic optical slit image width isThe incident and emergent optical pupil widths are $A_{\mathrm{coll}}=W\cos \alpha$ and $A_{\mathrm{cam}}=W\cos \beta$. Multiplying cancels the anamorphic magnification of a grating:This is conservation of the Lagrange optical invariant in the dispersion plane. It uses paraxial optical slit angles, an unvignetted optical pupil and matching full-width conventions for $\theta ,p,D,A_{\mathrm{cam}}$.

The number of spaxels is $M=(\beta /\theta {)}^2$. To obtain definite scaling exponents, keep the spectral resolving power, wavelength interval, diffraction grating angles and groove spacing, input focal ratio, and photodetector sampling of a spectral resolution element fixed. A fixed photodetector then accommodates a fixed number $M_0$ of spectra, so $N\simeq M/M_0$ (rounded up in an actual instrument). The corresponding etendue per spaxel scales as ${\theta }^2$. Angular ratios are independent of whether both angles are in arcseconds; optical invariant equations use radians.

Write $b$ for the collimated beam diameter. The diffraction grating result $R\theta D=W(\sin \alpha +\sin \beta )$ gives $b\propto W\propto \theta$. A fixed collimator focal ratio gives $f_{\mathrm{coll}}\propto b\propto \theta$. With a fixed image width $p$ in photodetector detector pixels, the invariant $\theta D=p{A}_{\mathrm{cam}}/f_{\mathrm{cam}}$ and $A_{\mathrm{cam}}\propto \theta$ instead imply $f_{\mathrm{cam}}$ is constant: the optical camera focal ratio increases as $1/\theta$.

In a simple on-axis beam-envelope model, each collimator and collimated disperser space has area proportional to $b^2$ and length proportional to $b$, whereas the optical camera cone has area proportional to $b^2$ and fixed length. HenceThe constants include $1/M_0$ and fixed design parameters. At fixed field size, the combined collimator and diffraction grating volumes scale as $M{\theta }^3\propto \theta \propto M^{-1/2}$, while the combined optical camera beam-cone volume scales as $M{\theta }^2\propto 1$.

These are integral-field spectrograph volume scaling laws for the shrinking beam, not a claim that the whole apparatus can shrink without a floor. A photodetector of fixed transverse size $d_0$ needs space for its field: an optical camera envelope interpolating between optical pupil width $b$ and photodetector width $d_0$ has volume proportional to $f_{\mathrm{cam}}(b^2+b{d}_0+d_0^2)$, rather than just $f_{\mathrm{cam}}{b}^2$. Replication then adds terms scaling as $M\theta$ and $M$, which can grow as the spaxels shrink. Other field clearances and mechanical margins also change the asymptote. Without the fixed spectral/design assumptions above, the information in the question does not determine unique physical-volume exponents. The role of fixed photodetector size and beam spread is discussed in Allington-Smith's instrument scaling model.

The optical pupil mapped onto the deformable mirror has actuator pitch approximately $s=D/N$. A sinusoidal wavefront error must have at least two actuator intervals per period to avoid aliasing, by the Nyquist–Shannon sampling theorem. ThusIf the $N$ actuator centres include both endpoints of the optical pupil diameter, the literal pitch is $D/(N-1)$ and the corresponding expression is $2D/(N-1)$; the paper uses the customary large-$N$ pitch convention. At the exact Nyquist boundary one quadrature can be poorly sampled, and real influence functions and actuator coupling reduce useful correction near the cutoff. The result is an ideal sampling bound, not a guarantee of arbitrary correction at its endpoint.

A point spread function is an imaging system's response to an optical point source. Here it means the distribution of image light intensity: an ideal point object becomes a finite spot, often with faint rings, because of diffraction. An unresolved star and a sufficiently small bead emitting fluorescence are practical approximations to an optical point source. Their image tells us how nearby object details will overlap, and reveals blurring from optical aberrations, defocus, and, for an optical telescope, astronomical seeing.

A useful convention separates shape from throughput. At a fixed image plane, let $p(\mathbf{x})\ge 0$ be the point spread function with ${\int }_{{\mathbb{R}}^2}p(\mathbf{x})d^2\mathbf{x}=1$. Its units are inverse area. A point delivering detected radiant flux $F$ at image position ${\mathbf{x}}_0$ produces light intensity $Fp(\mathbf{x}-{\mathbf{x}}_0)$; this point spread function normalization makes the integral over the image equal to $F$. A profile divided by its own peak is convenient for comparing widths, but is a different point spread function normalization and does not by itself compare total radiant flux.

In incoherent imaging, different object points have negligible mutual spatial optical coherence over the measurement, so their image light intensities add. If the response also obeys translation invariance over the chosen field of view, the expected continuous image is the convolutionHere $O$ is the object brightness distribution expressed in image-scaled coordinates, with magnification already included, and $\tau$ is constant transmission. For $O(\mathbf{u})=F_0\delta (\mathbf{u}-{\mathbf{u}}_0)$, where $\delta$ is the two-dimensional Dirac delta function, this gives $I=\tau F_{0}p(\mathbf{x}-{\mathbf{u}}_0)$. A changing response across the field of view instead requires a position-dependent integral kernel $p(\mathbf{x};\mathbf{u})$; one measured point spread function is then only a local calibration.

In coherent imaging, the superposition principle applies to the complex wave amplitudes. With amplitude spread function $h$, the corresponding ideal model isThus $|h\ast E{|}^2$ generally differs from $|h{|}^2\ast |E{|}^2$: coherent contributions produce wave interference terms that depend on their relative wave phases. Their time average vanishes for mutually incoherent emitters, which explains the intensity convolution. Partial optical coherence requires the correlations between object points as well.^[ref]

The figure is an original diffraction model, not measured data. All panels have the same total radiant flux: each member of a pair contributes half the single-source radiant flux. Dashed curves show the separate light intensity contributions; the solid curve is their sum. The point spread functions have tails, so even sources satisfying the Rayleigh criterion overlap. As separation decreases, the image changes continuously; there is no abrupt disappearance of all information at one separation.

A motivating measured example is Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others (2005). A $543.8\mathrm{nm}$ laser, delivered through an optical fiber, excited fluorescent beads on a slide through a $100\times$, numerical aperture $1.3$ oil-immersion objective. A moving stage scanned the bead; emitted light returned through the objective, an emission filter and a pinhole to a photodiode. The readout was fluorescence light intensity against scanned position. Under the same setup, nominal bead diameters $0.17$, $0.5$, and $1\mu \mathrm{m}$ produced progressively broader measured spots. The size comparison controls for the finite object: the measured response is the convolution of bead emission with the effective point spread function, rather than a pure point response. Inverting this convolution model gave a response compared with the theoretical one. The smaller bead gave a weaker signal relative to noise; uncertainty in actual bead diameter also affected the restored width. These are limits of the calibration. In confocal microscopy, excitation and pinhole detection jointly determine the effective point spread function; this experiment does not measure the ideal widefield Airy disk plotted above.^[ref]

For a new calibration, useful additional controls are to measure detector background, avoid saturating the photodetector, repeat the scan to check stability, and compare isolated beads at several positions in the field of view. Record wavelength, numerical aperture, pinhole setting, sampling interval and optical focus. These are practical recommendations, not extra measurements claimed for the published experiment.

Take $N=4p-1$, $|A|=|B|=2p-1$ and sign vectors $u_A=21_A-1$. The Frobenius inner product of two outer products is $(u_A\cdot u_B{)}^2$, proving the formula. The squared Euclidean distance is largest precisely when $|A\cap B|=p-1$, where the inner product is minus one. The common diagonal can be deleted, leaving an isometric representation in dimension $\left(\genfrac{}{}{0ex}{}{N}{2}\right)$ if off-diagonal coordinates are multiplied by $\sqrt{2}$. The Frankl-Wilson theorem then bounds each smaller-diameter class.

A spatial resolution element is one independently resolved angular or spatial scale under a specified resolution criterion. For an ideal telescope of diameter $D$, its angular scale is of order $\lambda /D$ and its angular area of order $(\lambda /D{)}^2$. A photodetector detector pixel or spaxel can be smaller than a resolution element without adding independent spatial information.

On $\{0,1{\}}^{\mathbb{N}}$ this sum is a metric, since each summand obeys the triangle inequality and a zero sum forces coordinatewise equality. A prefix cylinder set fixing the first $n$ coordinates contains the open ball of radius $2^{-n}$ about each of its points and has diameter at most $2^{-n}$. These bounds show that the metric topology equals the prefix cylinder set topology, the usual Cantor space topology.